Modelling Rayleigh–Taylor instability of a sedimenting suspension of several thousand circular particles in a direct numerical simulation

In this paper we study the sedimentation of several thousand circular particles in two dimensions using the method of distributed Lagrange multipliers for solid–liquid flow. The simulation gives rise to fingering which resembles Rayleigh–Taylor instabilities. The waves have a well-defined wavelength and growth rate which can be modelled as a conventional Rayleigh–Taylor instability of heavy fluid above light. The heavy fluid is modelled as a composite solid–liquid fluid with an effective composite density and viscosity. Surface tension cannot enter this problem and the characteristic shortwave instability is regularized by the viscosity of the solid–liquid dispersion. The dynamics of the Rayleigh–Taylor instability are studied using viscous potential flow, generalizing work of Joseph, Belanger & Beavers (1999) to a rectangular domain bounded by solid walls; an exact solution is obtained.